Mutational robustness can facilitate adaptation

Mutational robustness can facilitate adaptationMutational robustness can facilitate adaptation Jeremy A. Draghi1, Todd L. Parsons1, Gunter P. Wagner3 & Joshua B. Plotkin1,2
Robustness seems to be the opposite of evolvability. If phenotypes are robust against mutation, we might expect that a population will have difficulty adapting to an environmental change, as several studies have suggested1–4. However, other studies contend that robust organisms are more adaptable5–8. A quantitative understanding of the relationship between robustness and evolvability will help resolve these conflicting reports and will clarify outstanding problems in molecular and experimental evolution, evolutionary developmental biology and protein engineering. Here we demonstrate, using a general population genetics model, that mutational robustness can either impede or facilitate adaptation, depending on the population size, the mutation rate and the structure of the fitness landscape. In particular, neutral diversity in a robust population can accelerate adaptation as long as the number of phenotypes accessible to an individual by mutation is smaller than the total number of phenotypes in the fitness landscape. These results provide a quantitative resolution to a significant ambiguity in evolutionary theory.
The relationship between robustness and evolvability is complex because robust populations harboura large diversity ofneutralgenotypes that may be important in adaptation9–11. Although neutral mutations do not change an organism’s phenotype, they may nevertheless have epistatic consequences for the phenotypic effects of subsequent muta- tions12–18. In particular, a neutral mutation can alter an individual’s ‘phenotypic neighbourhood’, that is, the set of distinct phenotypes that the individual can access through a further mutation. Pioneering studies based on RNA folding and network dynamics suggest that genotypes expressing a particular phenotype are often linked by neutral mutations intoa largeneutralnetwork,andthatmembersofaneutralnetworkdiffer widely in their phenotypic neighbourhoods1,19–21. Numerous studies have documented the importance of neutral variation in allowing a population to access adaptive phenotypes5,17,18,22–24, and neutral networks have consequently been proposed to facilitate adaptation9–11.
Here we analyse the relationship between robustness and evolvability using a population genetics model that specifies statistical properties of the fitness landscape. Our approach bypasses the tremendous complexity of explicit neutral networks1,11,17,19,21,22 to focus instead on the essential evolutionary consequences of epistatic mutations. We consider a population of N individuals reproducing according to the discrete-time, infinite-sites Moran model. In each time step, a randomly chosen individual produces one offspring. Upon replication, a mutation occurs with probability m, producing a novel genotype. With probability q, the mutation is neutral. The parameter q therefore quantifies robustness, which is assumed to be the same for all genotypes on a network. With probability 1 2 q, the mutation is non-neutral and changes the offspring’s phenotype to one of K phenotypes accessible from a given genotype. Each genotype has a specific set of K accessible phenotypes that constitute its phenotypic neighbourhood; these K phenotypes are drawn uniformly from P possible alternatives. Pheno- typic neighbourhoods are assumed to be independent, such that the K accessible phenotypes are redrawn whenever a mutation occurs (we
relax this and other assumptions below). When the number of phenotypes accessible to an individual, K, is significantly smaller the total number of alternative phenotypes in the landscape, P, neutral mutations can profoundly alter an individual’s phenotypic neighbourhood. This genotype–phenotype map is illustrated in Fig. 1.
Our model implicitly represents a space of adjacent neutral networks. Neutral mutations produce other genotypes on the focal network, whereas non-neutral mutations produce genotypes on adjacent networks, each expressing one of P alternative phenotypes. To study evolution on the focal network, we assume that initially all of the P alternative phenotypes are lethal (our results hold more generally; see Supplemen- tary Information, section 5). We analyse the relationship between robustness, q, and the time required to adapt to a novel environment; this analysis is outlined in Box 1 and detailed in Supplementary Information, section 1.
We find that a robust population may adapt either more slowly or more quickly than one that is less robust (Fig. 2). Starting from a steady-state population with robustness q, we consider an environmental shift that assigns one of the P alternative phenotypes the greatest fitness. We have derived an analytic expression for the mean waiting time before this fittest phenotype subsequently arises in the population (Supplementary Information, section 1.4). When all phenotypes are accessible from any genotype (K 5 P), neutral mutations have no epistatic consequences and we observe what is naively expected: more robust populations always adapt more slowly (Fig. 2). However, when the phenotypic neighbourhood size, K, is smaller than the total number of phenotypes, P, we find an unexpected pattern: the relationship between robustness and evolvability is non-monotonic. In particular, populations with an intermediate
1Department of Biology, 2Program in Applied Mathematics and Computational Science, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA. 3Department of Ecology and Evolutionary Biology, Yale University, New Haven, Connecticut 06520, USA.
Figure 1 | The genotype–phenotype model. Schematic representation of the genotype–phenotype map used in our analysis. Each circle corresponds to a genotype; colours denote phenotypes. The model parameter q quantifies robustness: a proportion q of mutations are neutral (solid lines) and the remaining mutations are non-neutral (dashed lines). A non-neutral mutation changes an individual’s phenotype to one of the K accessible alternatives that form the individual’s phenotypic neighbourhood. When K is smaller than the total number of alternative phenotypes in the landscape, P, individuals may have different phenotypic neighbourhoods. The central pair of adjacent genotypes shown here express the same phenotype, but they have different phenotypic neighbourhoods.
Vol 463 | 21 January 2010 | doi:10.1038/nature08694
353 Macmillan Publishers Limited. All rights reserved©2010
amount of robustness adapt more quickly than populations with little or no robustness (Fig. 2).
There is a simple explanation for this counterintuitive result. In a population with little robustness (small q), most mutations are lethal and little genetic variation accumulates. As a result, the population may not contain any adaptable individuals, that is, those that are a single mutation away from the beneficial phenotype. Thus, when q is small the population may need to wait a long time before an adaptable individual arises, and then wait further for the adaptive phenotype to arise. However, slightly more robust populations contain a greater diversity of neutral genotypes, each of which has an independent chance (probability K/P) of being adaptable; thus, more robust populations may adapt more quickly.
Adaptation is most rapid when a population has an intermediate level of robustness. Moreover, this optimal level of robustness increases as the ratio K/P decreases (Fig. 2). This trend confirms the primary intuition behind our result: when phenotypic neighbourhoods are small, less robust populations contain few individuals who are ‘prepared to adapt’. In this range (q is small and K , P), increasing robustness results in a larger repertoire of phenotypes accessible to the population, thereby accelerating adaptive evolution.
In addition to adaptation time, we have also studied another measure of evolvability, namely the diversity of phenotypes produced by mutations in a population in a steady state. Again, the naive expectation is that as robustness increases, fewer non-neutral mutants are produced each generation and, as a result, the diversity of mutant phenotypes should decrease. However, an increase in robustness also increases the neutral genetic diversity within a population, and when K is less than P, each additional neutral type may increase the number of phenotypes accessible to the population through mutation. Thus, as with adaptation time, an unexpected, non-monotonic, relationship is apparent when K , P: more robust populations can produce greater phenotypic diversity than their less robust equivalents (Fig. 3). We have derived an analytic expression to quantify the range of parameters for which this relationship is non-monotonic (Supplementary Information, section 2). Our analysis shows that when K is smaller than a threshold determined by P, N and m, the diversity of mutant phenotypes is maxi- mized at an intermediate level of robustness.
There is an interesting difference between adaptation times and phenotypic diversity: increasing the population size or mutation rate makes the relationship between robustness and adaptation time more like the naive monotonic prediction, whereas it makes the relationship between robustness and phenotypic diversity less like the naive monotonic prediction (Supplementary Information, section 4). Although these influences of population size and mutation rate have some intuitive basis, they demonstrate that even qualitative predictions about the robustness–adaptability relationship require an explicit population genetics model.
Our analysis relies on four strong assumptions: a neutral mutation completely redraws the phenotypic neighbourhood; the number of phenotypes, K, in a genotype’s neighbourhood is independent of its robustness, q; the values of K and q do not vary across the neutral network; and alternative phenotypes are generally lethal. Relaxing each of these assumptions does not change our qualitative results (Supplementary Information, section 5). Briefly, we relax the first assumption by introducing a parameter, f, which is the fraction of K neighbours that are redrawn following a neutral mutation. Allowing correlations between the phenotypic neighbourhoods of neutral neighbours (that is, allowing f , 1) still preserves the non- monotonic relationship between robustness and evolvability. Furthermore, a strong linear correlation between K and q, or variation in either quantity across the network, does not change our results. When q varies across the network, the population evolves
Box 1 | Analysis of adaptation time
We study the time, following an environmental change, until the newly beneficial phenotype arises in a population with robustness q. A genotype is said to be ‘adaptable’ if its phenotypic neighbourhood contains the beneficial phenotype; our analysis links the stochastic evolution of these adaptable types to the adaptation time. Let p(t, y) denote the probability density of there being y adaptable individuals at time t, scaling space and time by the factor
ffiffiffiffi N p
. Then p(t, y) is well approximated by the solution to
Lp
Lt ~
L2
Ly
bKq
K yp(t, y)
where b 5 Nm. The first term in this expression quantifies genetic drift, the second term quantifies the increase in adaptable individuals through mutation and the third term describes the rate of mutations that produce the beneficial phenotype. The conflicting effects of robustness on adaptation are evident in this expression: an increase in robustness (q) increases the supply of adaptable individuals, but it also reduces the rate at which beneficial mutations arise in such individuals. Solving a boundary-value problem related to this equation produces an analytic expression for the expected arrival time of the beneficial phenotype (Supplementary Information,section 1), a graph of which is shown in Fig. 2.
Robustness, q
M ea
n ad
ap ta
tio n
tim e
(g en
er at
io ns
50
10
2,000
1,000
500
100
Figure 2 | Robustness and adaptation time. The relationship between robustness, q, and the average waiting time before the arrival of a specific beneficial mutation, for three fitness landscapes. Points show the means of 10,000 replicate Monte Carlo simulations, and lines show our analytic predictions (Box 1 and Supplementary Information, section 1). When all possible phenotypes in the landscape are directly accessible by a mutation from any genotype (that is, when K 5 P), robustness always inhibits adaptation (red curve). However, when phenotypic neighbourhoods are small (that is, when K , P), neutral mutations have epistatic consequences and the resulting relationship between robustness and adaptation time is non-monotonic: adaptation is most rapid at intermediate levels of robustness. N 5 10,000, m 5 0.001.
0.0 0.2 0.4 0.6 0.8 1.0
0
2
4
6
8
10
K = 5, P = 100
Figure 3 | Robustness and diversity. The relationship between robustness, q, and the diversity of phenotypes produced by mutation in each generation, for two fitness landscapes. Points show the means of 100,000 replicate simulations; arrows depict slopes calculated analytically (Supplementary Information, section 2). As these results demonstrate, an increase in robustness can increase phenotypic diversity, but only when the level of robustness, q, is small and the number of phenotypes accessible from a single genotype, K, is less than the total number of phenotypes in the landscape, P. N 5 10,000, m 5 0.001.
LETTERS NATURE | Vol 463 | 21 January 2010
towards greater robustness as predicted by previous studies25,26. Nonetheless, the time required to acquire a new adaptive phenotype is still accurately described by our analytic formula, replacing the fixed value of q by the average q in the population. The same relationship between robustness and adaptation also holds when alternative phenotypes are moderately deleterious, as opposed to lethal. Therefore, our conclusions are not sensitive to any of the strong assumptions used to derive our analytical results.
Our results reveal a complex relationship between robustness and evolvability. In some situations, increasing robustness will decrease evolvability, whereas in other situations it will accelerate adaptation. The latter phenomenon can occur only when the number of phenotypes accessible to an individual, K, is smaller than the total number of alternative phenotypes in the landscape, P. To assess the plausibility of this condition, and to test the assumptions and predictions of our abstract model using an empirical, mechanistic genotype–phenotype map, we examined the folding and evolution of simulated RNA molecules, using the Vienna RNA Package (version 1.6.1) to estimate reasonable values of K, P, and f for RNA. Because these parameters vary among genotypes in an RNA neutral network, we determined appropriate averages of K, P, and f (Supplementary Information, section 6.1). For sequences of length 40 nucleotides, we estimated that K < 19 and that P . 60,000, confirming that K , P for RNA. Furthermore, we found that f < 0.3, indicating that neutral mutations substantially alter phenotypic neighbourhoods. Finally, we evolved RNA populations in silico with varying levels of robustness, and observed a non-monotonic relation between evolvability and robustness, which was predicted accurately by our abstract model (Sup- plementary Information, section 6.2).
Recent studies have used theoretical27,28 or biological5,8 examples to argue that robustness increases evolvability. Another study has argued that robustness can either increase or decrease evolvability, depending upon the level at which robustness is described11. Although that study provided important intuition, it did not quantify the effects of robustness on adaptation in an evolving population. By contrast, our analysis describes the population genetics connecting these important properties. This perspective allows a quantitative resolution to opposing informal arguments, and highlights the complex interplay of influences shaping mutational robustness29,30.
Our analysis also reveals general patterns that may guide future experimental studies. First, the relationship between robustness and evolvability can be non-monotonic. In light of this complexity, empirical studies must go beyond pairwise comparisons of high- and low-robustness strains8, to measure evolvability over a broad range of robustness values. Second, the population size and mutation rate in part determine whether robustness increases or decreases adaptation time. This insight was not apparent from informal arguments linking robustness and evolvability9–11, and has not yet been considered in any empirical work. Finally, the parameters K, P, and f provide a new way to quantify epistasis beyond the conventional framework of synergistic and antagonistic interactions among selected sites.
Even though most standing genetic variation is neutral, the epistatic consequences of neutral mutations have received little experimental study. Our results demonstrate that conditionally neutral mutations strongly influence a population’s capacity to adapt; this form of ‘neutral epistasis’ therefore deserves direct experimental interrogation.
Received 4 November; accepted 19 November 2009.
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Supplementary Information is linked to the online version of the paper at www.nature.com/nature.
Acknowledgements We thank P. Turner and members of the Plotkin laboratory for advice and feedback. J.B.P. acknowledges funding from the Burroughs Wellcome Fund, the David and Lucile Packard Foundation, the James S. McDonnell Foundation, the Alfred P. Sloan Foundation, the Defense Advanced Research Projects Agency (HR0011-05-1-0057) and the US National Institute of Allergy and Infectious Diseases (2U54AI057168). G.P.W. acknowledges funding from the John Templeton Foundation and the Perinatology Research Branch of the US National Institutes of Health.
Author Contributions J.A.D., J.B.P. and G.P.W. designed the project. J.A.D. and J.B.P. wrote the paper; G.P.W. and T.L.P. edited the paper. J.A.D. performed the simulations. T.L.P. performed most of the analysis, with contributions from J.B.P. and J.A.D. J.A.D., T.L.P. and J.B.P. wrote the Supplementary Information.
Author Information Reprints and permissions information is available at www.nature.com/reprints. The authors declare no competing financial interests. Correspondence and requests for materials should be addressed to J.B.P. ([email protected]).
NATURE | Vol 463 | 21 January 2010 LETTERS
doi: 10.1038/nature08694Mutational robustness can facilitate adaptation Supplementary Information
Jeremy A. Draghi1, Todd L. Parsons1, Gunter P. Wagner3, Joshua B. Plotkin1,2
1Department of Biology, University of Pennsylvania, Philadelphia, PA, 19104, USA 2Program in Applied Mathematics and Computational Science, University of Penn- sylvania, Philadelphia, PA, 19104, USA 3Department of Ecology & Evolutionary Biology, Yale University, New Haven, CT, 06520, USA
1 Robustness and adaptation time
In this section we study how robustness is related to adaptation time.
1.1 Model Description
As in the main text, we consider a population of N individuals under the infinite alleles Moran model. In each discrete time step, a randomly-chosen individual produces one offspring, which replaces a random individual. A mutation occurs with probability µ and produces a unique genotype. With probability q a mutation is neutral; q therefore quantifies robustness. Otherwise, the mutation is non-neutral and changes the phenotype to one of K phenotypes accessible from a given genotype. Each genotype has a specific set of K accessible phenotypes which constitute its phenotypic neighborhood; these K phenotypes are drawn uniformly from P possible alternatives. Genotypes have independent phenotypic neighborhoods, so the K accessible phenotypes are redrawn whenever a mutation occurs. The model is described by the five parameters K, P , µ, q, and N .
Starting from a population in steady state, we consider an environmental shift that assigns one of the P alternative phenotypes the highest fitness. Before the environmental shift, we assume that all P of the alternative phenotypes are inviable, such that only genotypes expressing the wild-type phenotype survive and reproduce.
1
We consider a population evolving in this regime of stabilizing selection until it reaches steady-state, as described in Section 1.4 below. Sometime after the population reaches steady-state, the selective environment shifts such that one of the P alternative phenotypes is no longer inviable, but is instead more fit than the wild- type. In this section we derive an analytic expression for the mean adaptation time – i.e. the average amount of time elapsed before the newly beneficial phenotype first arises in the population.
Let t = 0 denote the time at which the alternative phenotype becomes more fit than the current phenotype – i.e. the time of the environmental shift. At this time we classify all genotypes in the population into three distinct groups. Genotypes that express the newly beneficial phenotype are called class C. Genotypes that express the wild-time phenotype are divided into two classes: those that can reach the optimal phenotype by a single point mutation, called class B or ’adaptable’, and those that cannot reach the optimal phenotype by a single mutation, called class A. This simplification into three classes is equivalent to the infinite-allele model described in the main text because all genotypes within a class have identical mutations rates to other classes.
A mutation that arises in a genotype of class A produces a genotype of class B with probability qK
P , and it produces another genotype in class A with probability
q(1 − K P ). The same holds for mutations from class B to class A. However, a
mutation arising in a genotype of class B might alternatively produce a genotype of class C; this probability of this occurrence is given by the chance that the mutation is non-neutral, (1− q), times the probability that a non-neutral mutation will produce the distinguished beneficial phenotype out of K phenotypic neighbors, 1
K . These
mutation rates are summarized in Supplementary Figure 1. We assume that the population dynamics follow a discrete-time Moran model
with a total population size N : at each time step, a randomly-chosen individual produces one offspring that replaces any individual, including the parent, with equal probability. We assume that all individuals are equally likely to give birth, though our results remain unchanged if we assume one of class A or B is weakly selected.
Aµ(1−K P )q
Supplementary Figure 1 – Probability of mutation within and between genotype classes
2
2www.nature.com/nature
doi: 10.1038/nature08694 SUPPLEMENTARY INFORMATION
We consider a population evolving in this regime of stabilizing selection until it reaches steady-state, as described in Section 1.4 below. Sometime after the population reaches steady-state, the selective environment shifts such that one of the P alternative phenotypes is no longer inviable, but is instead more fit than the wild- type. In this section we derive an analytic expression for the mean adaptation time – i.e. the average amount of time elapsed before the newly beneficial phenotype first arises in the population.
Let t = 0 denote the time at which the alternative phenotype becomes more fit than the current phenotype – i.e. the time of the environmental shift. At this time we classify all genotypes in the population into three distinct groups. Genotypes that express the newly beneficial phenotype are called class C. Genotypes that express the wild-time phenotype are divided into two classes: those that can reach the optimal phenotype by a single point mutation, called class B or ’adaptable’, and those that cannot reach the optimal phenotype by a single mutation, called class A. This simplification into three classes is equivalent to the infinite-allele model described in the main text because all genotypes within a class have identical mutations rates to other classes.
A mutation that arises in a genotype of class A produces a genotype of class B with probability qK
P , and it produces another genotype in class A with probability
q(1 − K P ). The same holds for mutations from class B to class A. However, a
mutation arising in a genotype of class B might alternatively produce a genotype of class C; this probability of this occurrence is given by the chance that the mutation is non-neutral, (1− q), times the probability that a non-neutral mutation will produce the distinguished beneficial phenotype out of K phenotypic neighbors, 1
K . These
mutation rates are summarized in Supplementary Figure 1. We assume that the population dynamics follow a discrete-time Moran model
with a total population size N : at each time step, a randomly-chosen individual produces one offspring that replaces any individual, including the parent, with equal probability. We assume that all individuals are equally likely to give birth, though our results remain unchanged if we assume one of class A or B is weakly selected.
Aµ(1−K P )q
Supplementary Figure 1 – Probability of mutation within and between genotype classes
2
SUPPLEMENTARY INFORMATIONdoi: 10.1038/nature08694
We are interested in the first time to arrival of an individual of type C. Prior to the first arrival, the total number of individuals is held fixed at N , and so it suffices to keep track of the number of individuals in class B until the first arrival. When the first individual of class C arises, we assume that the process jumps to an absorbing “graveyard” state , at which time we stop the process. We denote this Markov stopping time by τ. We wish to compute the expectation of τ in terms of the parameters K, P , µ, N , and q.
Let XN(t) denote the number of individuals in class B in the tth time step for t < τ. For t ≥ τ we let XN(t) = . Then XN(t) is a discrete-time Markov chain
on the set I def = {0, . . . , N} ∪ {}. The transition probabilities of this chain,
QN i,j = P {XN(t+ 1) = j|XN(t) = i} ,
are given by
i
N
i
N
i,,
for 0 ≤ i ≤ N . Furthermore, QN , = 1, and QN
i,j = 0 for all other pairs i, j ∈ I. We use the notation QN to denote the matrix with entries QN
i,j, the generator of the Markov chain XN(t).
1.2 Two continuous-time approximations
The discrete process XN(t) is too complicated to consider directly, so we will introduce two different limiting processes that asymptotically capture the essential behavior. We will use these continuous-time approximations to derive an analytic expression for the mean adaptation time. This approximation will be asymptotically accurate for large N .
We analyze two different time- and mass-rescaled processes:
Y1,N(t) = η−1 1,NXN (α1,N t) ,
Y2,N(t) = η−1 2,NXN (α2,N t)
3
where we define the choices of α1,N 1, η1,N 1, α2,N 1, and η2,N 1 below. For each n = 1, 2 note that Yn,N is a continuous-time processes, whereas XN is discrete. One unit of time for Yn,N(t) corresponds to αn,N time units in the discrete process XN(t). By definition, Yn,N(t) is O(1) when XN (αn,N t) ∝ ηn,N .
We show below that for small initial values XN(0), by choosing η1,N = N 1 2 and
α1,N = N 3 2 the process Y1,N(t) converges in distribution to a diffusion process with
killing, Y1(t), on R +
def = [0,∞) ∪ {}, with transition density function
p(t, z, y) dz = P{Y1(t) ∈ [z, z + dz)|Y1(0) = y},
satisfying the forward Kolmogorov equation
∂p
∂t =
∂2
K zp(t, z, y).
where β = Nµ. Thus, when the number of adaptable individuals is small, back mutations have negligible effect and the rate of mutation to the adaptable type is effectively constant. Changes in number are a result of three effects, encapsulated in the three terms: genetic drift, mutations from non-adaptable to adaptable types and finally mutations from adaptable types to the target phenotype. Even when the specific form of the mutation rates is varied, as we will consider below, this asymptotic form, and the interpretation of its three terms, remains unchanged.
We will also show below that by choosing η2,N = N and α2,N = N the process Y2,N(t) converges to a pure jump process that jumps into state with probability one, after a random time which is exponentially distributed with rate
β(1− q)
K Y2(0).
In the following sections, we give a proof of these two limits. Readers who are less interested in the formal details may skip ahead to Section 1.3 for the derivation of the expected first arrival time.
1.2.1 Generators of Stochastic Processes
Let Y (t) be some continuous-time stochastic process, and let Py and Ey denote probability and expectation conditioned on Y (0) = y, respectively. We recall from (Karlin & Taylor 1981, Ethier & Kurtz 2005) that the infinitesimal generator A of Y (t) is
Af(y) def = lim
for all f for which the limit on the right exists. We refer to all such f as the domain of A, D(A). For continuous time and continuous state processes, the generator plays a role analogous to the transition matrix, and may be used, in conjunction with restrictions on the domain, to uniquely characterize the process. Infinitesimal generators provide a convenient unifying framework within which to study Markov processes.
For example, for a diffusion process with killing Z(t), with probability transition density p(t, z, y), i.e.
Py {Z(t) ∈ (a, b)} = b
a
∂p
the corresponding generator is
The domain of A consists of all twice differentiable functions satisfying appropriate boundary conditions (absorbing, reflecting, etc.) and vanishing at the graveyard point .
A continuous time Markov jump process, where the time to the next jump, starting from y, is exponentially distributed with rate r(y), and µ(y, dz) is the probability that the process jumps from y to z, has generator
Af(y) = r(y)
A1f(y) = yf (y) + βKq
K yf(y). (1.1)
corresponding to ηn,N = N 1 2 , for ηn,N = N , we get generator
A2f(y) = β
= − β
for f continuous in [0, 1] (and vanishing at ). The latter generator describes a Markov jump process with jump rate r(y) = β
K (1− q)y and jump distribution given
by a Dirac point mass, µ(y, dz) = δ(dz), i.e. all jumps are to the graveyard state. A2 uniquely characterizes the corresponding process. For A1 however, we also need to determine appropriate boundary conditions, which we discuss below.
1.2.2 Domain and Boundary Conditions for A1
In general, the infinitesimal generator does not uniquely specify a diffusion process. We must also characterize the domain to which we apply the generator by deter- mining or imposing appropriate boundary conditions. We do so by following Feller’s boundary classification (Feller 1954a,b) (see also (Karlin & Taylor 1981, Ethier & Kurtz 2005)).
For A1,∞ is always a natural boundary, and cannot be reached in finite time. Ac- cording to Feller’s classification scheme we consider the parameter ν = 1
2
.
If ν < 0, then 0 is a regular boundary, and it is an entrance boundary if ν > 0. In the former case, the process can enter or leave at 0, and we may specify any boundary condition from absorption to reflection. In the latter, the process can enter the inter- val (0,∞) in finite time if started from 0, but can never reach 0 when started from an interior point. Since our original finite Markov chain model allows for mutation away from 0, we will impose reflecting boundary conditions at 0 for ν < 0.
Whether the boundary behavior at 0 is entrance or reflecting, the corresponding condition for a function f belonging to D(A) is the same,
lim y↓0
lim y→∞
f(y) = 0.
We also require that be an absorbing state, which implies that
f() = 0
for all f ∈ D(A1). Lastly, we require that all functions in D(A1) be continuous on [0,∞) and be twice continuously differentiable on (0,∞). Together, these conditions specify the domain of A1 and uniquely determine the process Y1(t).
6
1.2.3 Deriving the Limiting Generators
To prove convergence, Theorem 6.5 in Chapter 1 of (Ethier & Kurtz 2005) proves that it suffices to show that, for appropriate choices of αn,N and ηn,N ,
lim N→∞
for n = 1, 2 and f ∈ D(Ai) ∩ C3(0,∞), where
Gn,N = {η−1 n,Nj|j ∈ N, 1 ≤ j ≤ N},
is the set of all possible values of Yn,N(t). Now, for y = η−1
n,N i ∈ Gn,N ,
j∈I
QN i,jf(η
− αn,NQ
n,N i)
Taylor expanding f(y) about y = η−1 n,N i, and recalling that f() = 0, yields
= αn,N
1
3f (ξ)
η−1 n,Nf
η−2 n,Nf
η−3 n,Nf
−1 n,N i)
for some ξ between η−1 n,N i and η−1
n,Nj. Substituting y = η−1 n,N i and β = Nµ into QN
i,j
We look for scalings for which all terms are finite, so that the limit is well- posed, and for which the killing term is non-zero. For the former, we must have ηn,N = O(N
1 2 ) or smaller, while for the latter, we must have αn,N = η−1
n,NN 2.
If we take α1,N = N 3 2 and η1,N = N
1 2 , then
α1,N(QN − I)f(y)→ A1f(y)
as N →∞, giving the diffusion limit, while if we take η2,N = N and α2,N = N , then
α2,N(QN − I)f(y)→ A2f(y)
Scalings with N ηN N 1 2 give rise to a generator identical to A2,
Af(y) = − β
K (1− q)yf(y),
only the domain now consists of functions continuous on [0,∞) and vanishing at . All other choices of αN and ηN result in a generator that becomes unbounded or tends to 0 as N →∞.
1.3 Adaptation time from fixed initial condition
In this section we calculate the expected value of τ, the first time at which the newly beneficial phenotype arises, for each of our two limiting processes Y1(t) and Y2(t), conditioned on Y1(0) = y or Y2(0) = y:
T1(y) def = EY1
y [τ] .
Once we account for the rescaling of time and mass, each of these may be used as an asymptotic approximation to the expected first arrival time starting from a population with i individuals of class B, in our original discrete model:
TN(i) def = EXN
i [τ] .
In particular, for i ∝ N 1 2 , we will use Y1(t) to approximate XN(t), so
TN(i) ∼ N 3 2T1(N
− 1 2 i), (1.3)
while for i ∝ N , we will use Y2(t) to approximate XN(t), so
TN(i) ∼ NT2(N −1i). (1.4)
In fact, as we show in Section 1.4, Eq.(1.3) is valid for all 1 ≤ i ≤ N . When we consider generator A2 (Eq.(1.2)), the mean adaptation time is simply
the mean of an exponential distribution:
T2(y) = 1
. (1.5)
On this timescale, the numbers of individuals in class B remains fixed before C arrives. Unfortunately, this simple expression becomes unbounded as y → 0, because individuals in class C can only arise from B individuals. Moreover, the expression above cannot be integrated against the steady-state distribution (see below). Therefore, we must use a different time scaling in order to determine the expected arrival time starting from very small numbers in class B. We will use the diffusion approximation with generator (1.1).
Since is the only absorbing state for our process, T1(y) is the expected first time to absorption for Y1(t) conditioned on starting from y, which can be written as a Dirichlet problem for the generator (Karlin & Taylor 1981),
A1T1(y) = −1 T ∈ D(A1)
lim y→∞
T1(y) = 0.
This can be readily solved via the method of Green’s functions (Karlin & Taylor 1981).
We first find solutions u0(y), u∞(y) to the homogeneous problem A1u(y) = 0,
with u0(y) and u∞(y) satisfying the boundary conditions at zero ( u0 s (0+) = 0) and
infinity (u∞(∞) = 0), respectively:
where a =
1− βqK
. Iν(z) andKν(z) are the modified
Bessel functions (Abramowitz & Stegun 1965). The Green’s function G(y, ξ) is then given by
G(y, ξ) =
9
where m(y) = y−2ν .
Intuitively, G(y, ξ) dξ represents the expected time Y1(t) spends in [ξ, ξ + dξ), given that Y1(0) = y. Thus the expected adaptation time is given by
T1(y) =
y
0
ay 2
2 , (1.6)
where 1F2(a1; b1, b2; z) is a generalized hypergeometric function (Slater 1966). This equation, while unwieldy, gives an analytic expression for the mean adaptation time, starting from a fixed frequency of B-type individuals.
Using asymptotic properties of the Bessel functions, it is possible to show that
T1(y) = 1
, (1.7)
in agreement with (1.5), the expression we previously obtained for large initial frequencies. Thus (1.6) is valid not only for small initial frequencies of XN(0), but in fact gives a uniform asymptotic estimate for all starting frequencies XN(0).
1.3.1 Approximation near y = 0
From Abramowitz & Stegun (1965), we have
Iν(z) ∼ 1
Γ(1 + ν)
∞ y ξ−νKν(aξ) dξ and
y
0 ξ−νI−ν(aξ) dξ, and simplifying using the re-
flection formula for the gamma function (Abramowitz & Stegun 1965),
Γ(ν)Γ(1− ν) = π
sin(νπ) ,
we obtain an asymptotic expression for T1(y) for y very small,
T1(y) =
√ π
2a
Γ
+ o(y)
In particular, we obtain a simple expression for the expected first arrival time from a population consisting only of A-type individuals,
T1(0+) =
√ π
2a
Γ
.
We may further simplify this using Gauss’ duplication formula for the gamma function (Abramowitz & Stegun 1965),
Γ(2z) = 22z−1
1.4 Adaptation time from steady state
Although (1.6) gives an analytic expression for the mean adaptation time starting from a fixed initial number of class-B genotypes, XN(0), we are actually interested in the adaptation time starting from a population in steady state. Therefore, we must integrate (1.6) over the probability distribution for the frequency of class-B genotypes in steady state.
Prior to the environmental shift, we have assumed that all genotypes not expressing the wild-type, including those in class C, are inviable. Therefore, the relative frequencies of class A and class B genotypes follow a standard, neutral Moran model
11
with asymmetric mutations between two types. Genotypes of class A mutate to class B at rate βqK
P , and class B mutates to class A at rate βq
1− K
. Therefore the
steady state frequency of class B individuals follows the well-known beta distribution (Ewens 2004) with probability density function
P {XN(0) = x} = g(x) = Γ(βq)
Γ(β(1−K P )q)Γ(β(
K P )q−1.
Thus, the expected adaptation time, in generations, starting from a population at steady state is
T = N 1 2
g(x)T1(xN 1 2 ) dx. (1.11)
The integrand above is difficult to compute numerically for large arguments, x. For- tunately, for x large we have the asymptotic expansion given by Eq. 1.7. Therefore, in practice we numerically evaluate the expression above by dividing the integral into two regimes:
T ≈ N 1 2
3/ √ N
g(x)T2(x) dx
When K P , almost all of the mass of the Dirichlet distribution is concentrated near x = 0; i.e. with very high probability there are few or no adaptable types at time t = 0. Thus, some degree of robustness is necessary for the arrival of sufficient adaptable types to survive drift and mutate to the target phenotype. As q increases from 0, the rate of arrival of adapted phenotypes increases, accelerating the arrival of the target type. However, when q approaches 1, the rate of mutation to the target type becomes exceedingly small, creating the observed non-monotonicity in the expected arrival time.
2 The relationship between robustness and phe-
notypic diversity
In this section we analyze how robustness, q, is related to the diversity of phenotypes produced by mutation, each generation, in a population at steady state. This calculation avoids the temporal complexity of waiting times and addresses a simple, general question: can populations that are more robust to mutation ever produce greater phenotypic diversity?
As before, we assume that one phenotype is fit and that the P alternative phenotypes are lethal. Mutations to a given genotype can produce only K phenotypes,
12
q is the proportion of mutations that are neutral, and a neutral mutation produces a novel genotype with a new set of K mutational neighbors. Diversity is measured as the expected number of distinct lethal phenotypes produced in a single generation, which we refer to as D(q).
We wish to understand under what parameters a population with some level of robustness can produce more phenotypic diversity than a comparable population with no robustness in the genotype-phenotype map. In other words, we wish to
understand the derivative dD(q) dq
q=0 . In this section we derive an analytic expression
for this derivative in terms of the model parameters K, P , µ, and N . As before we consider a haploid population of N asexuals which mutate at a
rate µ and reproduce according to the Moran process. Let A denote the number of distinct neutral genotypes present in the population. We define
βq = Nµ(1− q)
1− µq .
βq denotes the expected number of non-neutral mutations that arise in the population, each generations. βn is the neutral mutation rate appropriate to the infinite- alleles Moran model (Ewens 2004), which we use to determine the limiting number and distribution of neutral alleles.
Let αA,m denote the expected number of novel phenotypes among m mutations arising in a population with A neutral types, given that each mutation gives rise to one of K equally-probable neighboring phenotypes. Then, the number of non-neutral mutations is Poisson distributed with mean βq, and
D(q) = ∞
αk,mP{A = k}. (2.1)
If q = 0, there there is only one neutral genotype present, i.e. A = 1 with probability one. This genotype can produce up to K alternative phenotypes by mutation. The expected number of non-neutral mutations in a single generation is then β0 = Nµ. We thus have
D(0) = ∞
We now turn to computing α1,m. Let ξi = 1 if the ith mutation gives rise to a novel phenotype, and 0 otherwise. Then,
ξ = m i=1
P{ξi = 1} = 1− 1
K
E [ξi] = K
K
m
We now turn to D(q) for q > 0. Since we are only interested in the derivative dD(q) dq
q=0 , we need only determine D(q) up to o(q). To this end, we observe that
βn = Nµq
= β0q +O(q2).
Next, we consider P{A = k}. The distribution of neutral genotypes in the infinite- alleles Moran model may be obtained via Hoppe’s urn (Ewens 2004). Briefly, we start with a population of size 1. At each time step, with probability i−1
i−1+βn a random
individual produces a clone, while with probability βn
i−1+βn , we add an individual with
a novel genotype. After N iterations, we have a population with N individuals, for which
P {A = k} =
.
i− 1 i− 1 + βn
= 1− β0qHN−1 +O(q2),
Hn ≡ n i=1
βn
= β0qHN−1 +O(q2),
(2.4)
while P {A = k} = O(qk−1) for k > 2. Thus, in the limit q → 0, we need only consider populations composed of one or two neutral genotypes.
We next turn to α2,m, the expected number of unique phenotypes given m mutations and A = 2. If pi is the probability that there are N − i individuals of the first genotype and i of the second, the expected number of unique phenotypes is:
α2,m = N−1 i=1
piγm,i, (2.5)
where γm,i is the expected number of phenotypes, resulting from m mutations in a population with (N − i) type 1 individuals and i type 2 individuals. Note that types 1 and 2 are equivalent (they both encode the wild-type phenotype) except that each has an independent set of K phenotypic neighbors.
We find γm,i as before: let ξj,k (j = 1, 2, k = 1, . . . , i) be 1 if the kth mutation to one of the type j individuals gives rise to a new phenotype. Without loss of generality, we first consider mutations in type 2. As above, the total number of mutations to type 1 is
ξ1 = i
k=1
ξ1,k
and E[ξ1] = α1,i. Now ξ2,k is 1 if and only if the k th mutation gives rise to different
phenotype from all k − 1 previous mutations to type 2 individuals, and moreover, that phenotype was not produced by a mutation to a type 1 individual. For the latter to be true, either the new phenotype is inaccessible to type 1, with probability 1− K
P , or the phenotype is adjacent to type 1, with probability K
P , but was not one
of the ξ1 phenotypes that arose. Thus,
P {ξ2,k = 1} = 1− 1
K
k=1 ξ2,k,
K
P
P ,
15
and, since the m mutations are binomially distributed among the two genotypes,
γm,i = m j=0
α1,jα1,m−j
P
. (2.6)
Finally, the probability that there are i individuals with the first genotype, given that A = 2, can be determined via Hoppe’s urn. At each step, a new individual is added. The first individual of the second genotype arrives at step k + 1 with probability
βn
1 βn+k
,
joining the k individuals of type 1 already present. Treating the k type 1’s and the single type 2 as separate lineages, the probability of i individuals of type 2 in the total population of N is simply the proportion of partitions of N with one partition of size i (Joyce & Tavare 1987),
N−i−1 k−1
N−1 k
pi =
1
N−1 k
+O(q) (2.8)
D(q) = ∞
+O(q2)
+O(q2)
while subtracting (2.2), dividing by q, and taking q → 0 yields
dD(q)
dq
m! [β0HN−1(α2,m − α1,m)− (m− β0)α1,m] . (2.9)
Supplementary Figure 2 plots Eq. (2.9) for P = 90, N = 300, and µ = 0.04 for a range of K. Also plotted are simulation results for D
q calculated at q = 0.0001,
for 100 million replicates for each point.
16
0 20
40 60
dD dq q=0
Supplementary Figure 2 – Predictions using Eq. 2.9 (line) compared with the means of 100 million replicate simulations at q = 0.0001 (circles).
3 Simulation Methods
Monte Carlo simulations were performed in two main ways. For the data in Figure 2 & 3 in the main text and Supplementary Figures 4, matrices of transition probabilities were pre-calculated and used to simulate individual replicates of a Markov chain. Starting distributions for a two-allele Moran model were calculated using Eq. 3.58 in (Ewens 2004). These simulations were validated by comparison to individual-based simulations of populations ofN genotypes. Similar individual-based simulations were also used to generate the data shown in Supplementary Figure 9.
Code for all simulations was written in C or C++ and compiled using gcc 4.0.1. The GSL libraries (v.1.9) were used for pseudorandom number generation, special functions, and probability distributions.
4 Quantitative effects of N and µ on the optimal
level of robustness
We used numerical methods to explore how the adaptation time from steady state, given by Eq. 1.11, and the mutational diversity, described above in Section 2, depend on the population parameters N and µ. The tables below show the approximate q
17
which minimizes the adaptation time, or maximizes the mutational diversity, for a given N and µ. Note that the effects of N and µ can be approximated by considering only their product, β, but that β affects our measurements of evolvability in qualitatively different ways: increasing β decreases the q which minimizes adaptation time, but increases the q which maximizes mutational diversity.
N µ β q of minimum time q of maximum diversity 5000 0.001 5 0.67 0.1 10000 0.0005 5 0.68 0.13 10000 0.001 10 0.66 0.17 10000 0.0015 15 0.64 0.19 15000 0.001 15 0.65 0.19 10000 0.003 30 0.58 0.2 30000 0.001 30 0.6 0.2
Table 1 – Effects of population size and mutation rate on the q that minimizes adaptation time, and the q that maximizes mutational diversity. P = 100 and K = 5.
5 Relaxation of model assumptions
We relax each of the four main assumptions underlying the model presented in the main text: (1) a single neutral mutation redraws completely the phenotypes neighboring a genotype; (2) the number of phenotypes, K, in a genotype’s mutational neighborhood is independent of its robustness, q; (3) K and q do not vary among the genotypes on a neutral network and; (4) alternative phenotypes are lethal (or adaptive).
To relax the first assumption we introduced a new parameter, f , defined as the fraction of the K phenotypes neighboring a genotype which are redrawn after a mutation. Our original model is therefore equivalent to this generalization when f = 1. Considering this new parameter in light of the analysis above, it is clear that the only effect of f is to scale the mutation rates between neutral genotypes that are or are not adaptable – i.e. class B and class A genotypes, respectively. The mutation rate from A to B is therefore µqf
K P
. Since decreasing f is equivalent to increasing
P , values of f < 1 actually broaden the range of parameters for which robustness and evolvability are positively correlated. This is illustrated in Supplementary Figure 3, which shows analytic predictions of adaptation times for a range of f .
18
Supplementary Figure 3 – The predicted mean time before the arrival of a beneficial mutation with varying f . N = 10, 000, µ = 0.001, K = 5; a) P = 15; b) P = 100. f = 1, bottom curves; f = 0.5, middle curves; and f = 0.1, top curves. Decreasing f increases adaptation time, but the shape of the relationship with robustness remains consistent.
The second assumption is an implicit description of the topology of neutral networks. In our original model, the fraction of mutations that are neutral (q) was independent of the phenotypic diversity of non-neutral mutations (K). We relax this assumption by allowing these two quantities to be correlated. As q increases, the number of mutational neighbors with distinct phenotypes might be expected to decrease, and so the richness of those phenotypic mutants, K, might also be expected to decrease. Therefore, in relaxing the second assumption we explored a linear, negative relationship between q and an effective value of K, denoted K(q):
K(q) = K(1− q) (5.1)
∂p
∂t =
∂2
K zp(t, z, y),
for which the expected arrival time, T1(y) still takes the form of Equation (1.6), given an appropriate choice of a and ν (see Section 1). This process also results in a non-monotonic relationship between the arrival time of the adapted phenotype and the robustness, q, but the mechanism is slightly different. Here, the rate of mutation from the adaptable types to the target phenotype is constant, while the rate of mutation to the adaptable class approaches 0 for q near 0 or 1, and therefore leads to a non-monotonic relationship between robustness and evolvability.
19

K decreases with increasing q K increases with increasing q Constant K
Supplementary Figure 4 – The mean time before arrival of a beneficial mutation, with either K constant or K correlated with q according to Eq. 5.1 or Eq. 5.2. N = 10000, µ = 0.001, P = 200, and K = 20. Points are means of 10,000 replicate simulations.
Supplementary Figure 4 shows the results of simulations whenK is negatively correlated with q according to Eq. 5.1. Comparing the open circles to the closed circles, which show numerical results for the same parameters and K independent of q, we note a quantitative difference for larger values of q. However, the non-monotonicity is preserved, suggesting that our qualitative conclusions are not sensitive even to a strong negative correlation between K and q.
For the sake of completeness, we also explored a positive correlation between q and K:
K(q) = Kq+ 1 (5.2)
Simulations in this regime produced qualitatively similar results; see Supplementary Figure 4.
To relax the third assumption, we performed simulations in which either K or q
20
were allowed to vary by genotype. (P is a property of an entire fitness landscape and so cannot vary with genotype). To vary K, we assigned each genotype in our neutral network a value of K drawn independently from a distribution. K is therefore effectively re-drawn following a neutral mutation. Supplementary Figure 5 shows the relationship between robustness, q, and mean adaptation time in three different situations: constant K = 5, K drawn from a Poisson distribution with mean 5, and K drawn from a shifted geometric distribution with mean 5 (distributions were truncated at K = P ). Populations were allowed to evolve before the environmental shift until the number of adaptable individuals, and the distribution of K, had equilibrated. As the figure demonstrates, our results are remarkably robust even when we allow K to vary across the neutral network. In fact, our analytic prediction for the mean adaptation time is highly accurate, after replacing the fixed value of K, in the original model, with the mean of the distribution from which K is drawn, in the extended model.
0.2 0.4 0.6 0.8
Fixed K Poisson−distributed K Geometric−distributed K
Supplementary Figure 5 – The mean time before arrival of a beneficial mutation, for constant K and variable K. N = 10000, µ = 0.001, P = 100, and mean K = 5. Points show the means of at least 4,000 replicate simulations.
21
As before, this can be explained by considering the rate of mutation into the family of adaptable genotypes and the rate of mutation from the adapted types to the target phenotype. We omit the calculations here, but it can be shown that
the rate of mutation into the adaptable class takes the form µq E[K] P
, while the
rate of mutation into the target phenotype is obtained by averaging β(1−q) K
over all individuals in the population. As before, these two rates determine the expected arrival time of the target phenotype.
We also relaxed the third assumption by allowing q to vary among genotypes in a neutral network. To do so, we performed simulations in which an offspring inherits its parent’s value of q, plus a random perturbation whenever a neutral mutation occurs. These perturbations are drawn from a Gaussian with standard deviation 0.1; if a perturbation would produce a q outside of [0, 1], then q is left unaltered. This method of mutating q allows robustness to evolve. Prior theory suggests that a population should evolve towards elevated robustness (higher q) in a fixed environment, because more robust individuals have an effective selective advantage of order µ (van Nimwegen et al. (1999), Forster et al. (2006)). This is indeed what we observe: the mean q in a population, denoted q, tends to increase over time in a fixed environment. Figure 6 shows this behavior by plotting the distribution of q across the ensemble of replicate simulations. At the beginning of a simulation, each individual is assigned a q from the uniform distribution, and q is therefore approximately normally distributed with a mean of 0.5. After each population evolves for 10,000 generations in a fixed environment, the ensemble distribution of q has shifted significantly towards larger values (Supplementary Figure 6), reflecting the evolution of robustness in a fixed environment.
After 10,000 generations, we assign one of the P alternative phenotypes the highest fitness, and we record the subsequent arrival time of the first such adaptive mutation. The relationship between q in a population at the time of this environmental shift and the subsequent time before the arrival of a beneficial mutation is summarized in Supplementary Figure 7. In this figure, the results of almost 300,000 replicates are binned according to q and plotted as grey points. The line illustrates analytical results results with fixed q, while black dots show comparable simulation results for fixed q. Although q may continue to evolve between the 10,000th generation and the eventual arrival of a beneficial mutant, we nonetheless find that q is an excellent predictor of adaptation time. In fact, our analytic formula for the mean adaptation time is highly accurate, after replacing the fixed value of q in the original model with observed mean q in the population. We also performed simulations in which we waited 100,000 generations before allowing beneficial mutations; these
22
0 10
00 0
20 00
0 30
00 0
40 00
0 10
00 0
20 00
0 30
00 0
40 00
0 50
00 0
Supplementary Figure 6 – (a) Histograms of q among nearly 300,000 replicates at generation zero (left) and generation 10,000 (right). Parameters: N = 10000, µ = 0.001, P = 100, and K = 5.
simulations produced very similar results. Thus, our results are robust with respect to significant variation in q across a neutral network and the associated evolution of q within a population.
For the sake of completeness, we also allowed f to vary across the neutral network. We assigned each genotype in our neutral network a value of f drawn independently from a distribution. f is therefore effectively re-drawn following a neutral mutation. Supplementary Figure 8 shows the relationship between robustness, q, and mean adaptation time in two different situations: constant f = 0.5, f drawn from a normal distribution with mean 0.5 and variance 0.1 (truncated at zero and one). Populations were allowed to evolve before the environmental shift until the number of adaptable individuals, and the distribution of f , had equilibrated. As the figure demonstrates, our results are robust even when we allow f to vary across the neutral network. In fact, our analytic prediction for the mean adaptation time is highly accurate, after replacing the fixed value of f , in the original model, with the mean of the distribution from which f is drawn, in the extended model.
Finally, we considered the effects of relaxing the fourth assumption. In our original model all mutations were either neutral or lethal, prior to the environmental shift. We simulated an alternative model in which the previously-lethal phenotypes
23
ut at
io n
Supplementary Figure 7 – Mean arrival time of the first beneficial mutant. The grey crosses depict means binned according to the value of q at the time of the environmental shift. The solid line shows comparable analytical results for fixed q, while the black circles show simulations for fixed q. N = 10000, µ = 0.001, P = 100, and K = 5.
were assigned fitness 1 − s, reflecting a disadvantage of size s compared the fitness of the focal phenotype. As before, individuals are chosen to reproduce in proportion to their fitnesses.
To describe mutation among the phenotypic classes in this more general model it is helpful to extend the notation used above. Genotypes of type C express the phenotype that will be beneficial after the environment changes, and they have fitness 1 − s prior to that shift. Genotypes of type B can produce type C mutants, and they can have fitness 1, which we will call Bfit, or 1 − s, which we denote by Bdel. Similarly, type A genotypes cannot produce type C, and they may either be Afit or Adel. Any mutation, whether neutral, deleterious, or beneficial, causes the set of K neighbors to be redrawn. Therefore, Adel-types may or may not be able to mutate back to Afit, depending on the phenotypic neighborhood of that type. Finally, we impose a necessary constraint on phenotypic neighbors: a mutant’s phenotypic neighborhood must contain its parent’s phenotype. So, if a type C genotype mutates
24
Supplementary Figure 8 – The mean time before arrival of a beneficial mutation, for constant f and variable f . N = 10000, µ = 0.001, P = 100, and mean f = 0.5. Points show the means of at least 4000 replicate simulations.
to a deleterious phenotype, it must become Bdel. We used individual-based Moran simulations to explore the time elapsed until
the first type-C individual arises. Each simulation began with an average proportion K P individuals of type Bfit, with the remaining of type Afit. Populations evolved for
30,000 generations before the environmental shift, at which time type C individuals become beneficial. 30,000 generations was determined to be sufficient for equilibrium by observing the distributions of Bfit, Bdel, and type C individuals across an ensemble of evolving populations. We recorded the arrival time of the beneficial type in terms of the number of generations after the environmental shift; if type C individuals were present at the environmental shift, this time was recorded as zero.
When alternative phenotypes are strongly deleterious (s ≈ 1) we expect to see little difference between the results of the original, lethal-mutation model and the new model. However, when s ≈ 0 we anticipate a monotonic, negative relationship between robustness and evolvability. When s = 0 non-neutral mutations have no cost, contribute to diversity in phenotypic neighborhoods, and generate type C individuals. Therefore, we ask how large must s be to match qualitatively the results of our original, lethal-mutation model.
Supplementary Figure 9(a) confirms that the relationship between robustness and evolvability is still non-monotonic when alternative phenotypes are moderately dele-
25
0.2 0.4 0.6 0.8 5
10 15
s = 0.02 s = 0.05 s = 0.3
Supplementary Figure 9 – Adaptation time and phenotypic diversity when alternative phenotypes are deleterious, but not lethal. N = 10, 000, µ = 0.001, P = 100, and K = 5 for all simulations shown. The dotted lines displays the means for the original model with lethal alternative phenotypes. (a) Mean adaptation time for three values of s, the fitness penalty of alternative phenotypes. Each point is the mean of 10,000 replicate simulations. (b) Mean number of unique mutant phenotypes. Each point is the mean of 5000 simulations.
terious. Similarly, we also used Moran simulations to measure phenotypic diversity in steady-state populations when the alternative phenotypes were deleterious but not lethal. These results, shown in Supplementary Figure 9(b), demonstrate that robustness and phenotypic diversity can also exhibit a non-monotonic relationship when alternative phenotypes are moderately disadvantageous. While the assumption that other phenotypes are lethal is mathematically convenient in our analyses, it can be relaxed without changing our qualitative results.
6 RNA Simulations
6.1 Measuring Epistasis in the RNA Landscape
We used the Vienna RNA package to calculate the minimum-free-energy secondary structures of RNA nucleotide sequences. The resulting genotype-phenotype map
26
was analyzed to estimate P, K, and f for a range of sequence lengths, L. While there are significant differences between the RNA landscapes and the type of abstract landscapes considered in our model, these estimated values provide some intuition about reasonable parameter values.
Our model assumes that out of P equally-likely phenotypes, K are equally- accessible from a given genotype. Since phenotypes are not evenly distributed in the RNA landscape, either globally or in the neighborhood of a single genotype, we must calculate some effective number of types for RNA. We note that, in our model, if we introduce two mutations into individuals of the same genotype, then the probability these mutation produce the same phenotype is 1
K . Similarly, if the mutations
occur in individuals with different genotypes, the probability of producing the same phenotype is 1
P . Since these probabilities have an intuitive connection to the roles
of K and P in our model, we define the effective K and P in the RNA landscape to yield the same probabilities. Additionally, some RNA sequences have the trivial, unfolded shape as their minimum free energy structure. We choose to regard this unfolded phenotype as inviable. We therefore define Ke as the inverse of the probability, pclone, that two viable phenotypes produced by non-neutral mutations in clones of the same genotype, are the same. Similarly, we define Pe as the inverse of the probability, prandom, that two viable phenotypes produced by non-neutral mutations in individuals with random, viable genotypes, are the same. Both pclone and prandom are measured by sampling random genotypes of length L with viable structures, and recording the phenotypes of random, non-neutral mutants of those genotypes; one hundred thousand genotypes were sampled for each L. Ke is therefore an average K across all genotypes in the landscape, while Pe is a lower bound on the number of accessible phenotypes; both are plotted in Supplementary Figure 10 for a range of L.
Estimating f is complicated by variation in K across the network. In keeping with the approach for Ke and Pe above, we relate f to the probability that two mutants have identical phenotypes. In our model, f determines the expected fraction of K neighbors that differ between two immediate neighbors on a neutral network, for P large. We therefore measure the probability, pneighbor, that a viable, non- neutral mutant of genotype x has the same phenotype as a viable, non-neutral mutant produced from a neutral neighbor of x. pneighbor can be related to f , pclone, and prandom as follows. First, note that we expect an adjacent pair of genotypes to have (1− f)K phenotypic neighbors in common, and fK neighbors drawn from the remaining P − (1 − f)K possible phenotypes. A given phenotype can appear only once in a set of K neighbors, so if a phenotype falls among the shared (1−f) portion of the neighborhoods, then it cannot fall among the unshared, redrawn f portion
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50 60
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50 0
20 00
10 00
0 50
00 0
(b)
Supplementary Figure 10 – Ke and Pe for Simulated RNA Molecules of Length L
of either genotype’s neighborhood. Therefore, we only need to consider the cases where both mutants produce phenotypes in the shared part of their neighborhoods, with probability (1 − f)2, or both produce phenotypes in the unshared portion of their neighborhoods, with probability f 2. In the former case, the probability that both mutants will have the same phenotype is 1
(1−f)Ke , or pclone
1−f . In the latter case,
each mutant can be one of P − (1 − f)K possible phenotypes; if K P then the probability that they are the same phenotype is approximately 1
Pe , or prandom. These
considerations yield the following relationship among pneighbor, pclone, prandom, and f :
pneighbor ≈ (1− f) pclone + f 2 prandom (6.1)
We use the equation above to define the effective value of f for the RNA landscape, after first measuring pneighbor, pclone, prandom. Supplementary Figure 11 shows values of fe estimated in this way, for several sequence lengths L. These estimates complement the findings of (Sumedha et al. 2007), who examined the overlap in phenotypic distance of related, but not neighboring, RNA genotypes.
Together these estimates for the RNA folding landscape suggest that the space of possible phenotypes is very rich compared to the phenotypic neighborhood of a single genotype – i.e. K P . Furthermore, the phenotypic neighborhoods of neutral neighbors differ significantly. These measurements in RNA confirm the central
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f e
Supplementary Figure 11 – fe for simulated RNA molecules of length L.
premise of our model: a neutral mutation can have profound, epistatic consequences for the phenotypes of subsequent mutations.
6.2 Robustness and Evolvability in Evolving RNA Popula- tions
To more directly test our abstract model using the RNA genotype-phenotype landscape, we performed evolutionary simulations with populations of RNA sequences. In these simulations, genotypes were 24-base-pair sequences and phenotypes were the minimum-free-energy structures of these sequences. For a set of simulations, we first chose M structures to designate as ‘high-fitness;’ these correspond to the single, high-fitness type in our abstract model. Adjusting M allows us to tune the ease of adapting to high fitness, much as we set the ratio K
P in our general model. For
each replicate, we chose genotypes at random until one was found that folds into a non-trivial structure which is not a high-fitness type. The chosen phenotype represents the fit wild-type; all other structures, except the high-fitness types, are then considered to be inviable. The chosen genotype is then used to found a population of N clones, which then evolve according to a Wright-Fisher, discrete-generation model until the first high-fitness phenotype arises. The ’adaptation time’ is defined as the
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number of generations which pass before a high-fitness type arises. Note that each replicate begins from a potentially unique genotype, but the high-fitness phenotype remain consistent throughout a set of replicates.
We measured the robustness, q, of each initial genotype as the frequency of point mutations that do not alter the phenotype, as we have done throughout our analysis. We used this initial q to predict adaptation times according to the analytical formulae derived above. For these simulations, the initial population was clonal, and so either all individuals were pre-adapted or none were pre-adapted. Thus, we obtained the expected adaptation time by suming expected adaptation times from each of these initial conditions, weighted by their probability:
P{none pre-adapted}TN(0) + P{all pre-adapted}TN(N)
∼ P{none pre-adapted}T1(0) + P{all pre-adapted}T2(1),
for large N , where T1(0) is given by (1.10) and T2(1) is obtained from (1.5). To calculate predicted waiting times as a function of q, we measured three properties of these initial genotypes corresponding to the three parameters of our general model: the probability that a genotype is pre-adapted, the average mutation rate of pre-adapted genotypes to the high-fitness types, and the probability that a genotype which is not pre-adapted has a pre-adapted neighbor. These measurements correspond to K
P , 1
K ,
and f in our abstract model. Although we estimated f above for RNA landscapes in general, we found that specific landscapes, determined by the particular set of M high-fitness types, have values of f which are more relevant to evolution of those landscape. Also, the mutation rate to high-fitness types declined monotonically with q in the RNA data, and so we used a linear regression of this relationship in our analytic predictions.
Supplementary Figure 12 shows the mean adaptation time as a function of robustness, q, for two sets of RNA simulations. In both cases, N = 500 and µ = 0.0002; M = 500 in the lower curve, and M = 200 in the upper curve. Note that the empirical relationship between robustness and evolvability is somewhat more complex and differs from our analytical prediction; however, the primary trend matches the U-shaped prediction of our general, population-genetic model. Moroever, our general model accurately predicts the scale of the waiting times in the RNA simulations, over several orders of magnitude.
Our analysis of the RNA genotype-phenotype map suggests that robustness can facilitate adaptation under some circumstances. However, Ancel & Fontana (2000) found that evolving RNA populations became less able to adapt as their robustness increased. While several differences between their study and ours may contribute to this discrepancy, the most fundamental one is that Ancel and Fontana did not
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Robustness, q
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Supplementary Figure 12 – The mean time before arrival of a beneficial mutation in RNA simulations. Each point is the mean of replicate simulations, and lines represent analytical predictions as described in the text. N = 500 and µ = 0.0002; M , the number of high-fitness phenotypes, is 500 in the lower curve, and 200 in the upper.
manipulate robustness to test its effect on adaptation. In fact, populations typically became more robust as they evolved, such that robustness and distance from the optimum were confounded. By specifying a set of high-fitness types, rather than evolving population toward a distant optimum, our simulations reduce the effect of confounding variables. We also note that Cowperthwaite et al. (2008) found no relation between the size of the neutral network on which a population began, and its success at evolving to a distant phenotypic optimum; similarly, Wagner (2008) found no relationship between the robustness of a sequence, and a measure of its evolvability. The major difference between our results and those negative findings is that we designate a set of structures as high-fitness, as opposed to picking a single, optimal structure. As a result, the populations evolved much more rapidly in our simulations, compared with the slow, step-wise process of adaptation to a distant optimum. The relative speed of adaptation in our simulations suggests that the robustness of the initial genotype has a much greater influence on adaptation than in previous studies.
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For all simulations we used version 1.6.1 of the Vienna package. Sequences were folded at 37C with the option ‘dangles’ set to ‘2.’ These settings produced the same landscapes as those described in (Cowperthwaite et al. 2008).
References
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Ancel, L. W. & Fontana, W. (2000), ‘Plasticity, evolvability, and modularity in rna.’, J Exp Zool 288(3), 242–283.
Cowperthwaite, M. C., Economo, E. P., Harcombe, W. R., Miller, E. L. & Meyers, L. A. (2008), ‘The ascent of the abundant: How mutational networks constrain evolution’, Plos Computational Biology 4(7).
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Sumedha, Martin, O. C. & Wagner, A. (2007), ‘New structural variation in evolutionary searches of rna neutral networks’, Biosystems 90(2), 475–485.
van Nimwegen, E., Crutchfield, J. & Huynen, M. (1999), ‘Neutral evolution of mutational robustness’, Proceedings of the National Academy of Sciences of the United States of America 96(17), 9716–9720.
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Title
Authors
Abstract
References
Figure 2 Robustness and adaptation time.
Figure 3 Robustness and diversity.
Box 1 Analysis of adaptation time

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Mutational robustness can facilitate adaptation